# Branched Coverings, Degenerations, and Related Topics 2022

## Speakers

Satoru Fukasawa (Yamagata)
Benoît Guerville-Ballé (Kyushu)
Ryosuke Masuya (Tokyo Metropolitan)
Masaru Nagaoka (Kyushu)
Hironobu Naoe (Chuo)
Takayuki Okuda (Hiroshima)
Sakumi Sugawara (Hokkaido)
Jun Ueki (Tokyo Denki)

## Schedule and links to abstracts

March 07(Mon)
10:00-11:00
Benoît Guerville-Ballé (Kyushu)
Construction of non-connected moduli space of line arrangements
11:20-12:20
Sakumi Sugawara (Hokkaido)
Divides with cusps and Kirby diagrams for line arrangements
14:00-15:00
Ryosuke Masuya (Tokyo Metropolitan)
The geometry of three sections on certain rational elliptic surfaces and Mumford representations
15:20-16:20
Hironobu Naoe (Chuo)
Presentation of the fundamental groups of complements of shadows
March 08(Tue)
10:00-11:00
Masaru Nagaoka (Kyushu)
Pathologies on log del Pezzo surfaces in characteristic five
11:20-12:20
Satoru Fukasawa (Yamagata)
Algebraic curves admitting the same Galois closure for two projections
14:00-15:00
Jun Ueki (Tokyo Denki)
Weber’s class number problem for cyclic covers of knots
15:20-16:20
Takayuki Okuda  (Hiroshima)
Representation theory and combinatorics on compact homogeneous spaces

## Abstracts of Talks

Speaker: Satoru Fukasawa (Yamagata)
Title: Algebraic curves admitting the same Galois closure for two projections
Abstract: The projections of plane curves from points of the projective plane and their Galois closures are considered. The following problem is natural: for different points, when are the Galois closures the same? As a main result of this talk, a criterion for the existence of a plane model of an algebraic curve such that the Galois closures of projections from two points are the same is presented, namely, the problem is settled. In general, it is hard to give an explicit description of the Galois closure of the projection from a point. Using the criterion, we present several examples of plane curves admitting two (non-Galois) points from which the Galois closures of projections are the same and are described explicitly. For example, the Hermitian function field in positive characteristic becomes the Galois closure of projections of a plane curve from two (non-uniform) points. This is a joint work with Kazuki Higashine (Yamagata) and Takeshi Takahashi (Niigata).
Speaker: Benoît Guerville-Ballé (Kyushu)
Title: Construction of non-connected moduli space of line arrangements
Abstract: We will present the construction of a splitting-polygon structure on a line arrangement. Then, we will see how such a structure allows to obtain non-connected moduli space. We will conclude with the reconstruction -using this method- of some classical examples of line arrangements with a non-connected moduli space (MacLane, Falk-Sturmels and Nazir-Yoshinaga), but also some new arrangements of 11 lines. If time permits, we will present potential continuations of this work.
Speaker: Ryosuke Masuya (Tokyo Metropolitan)
Title: The geometry of three sections on certain rational elliptic surfaces and Mumford representations
Abstract: We study the geometry of plane curves obtained by three sections and another section given as their sum on certain rational elliptic surfaces. We make use of Mumford representations of semi-reduced divisors in order to study the geometry of sections. As a result, we are able to give new proofs for some classical results on singular plane quartics and their bitangent lines.
Speaker: Masaru Nagaoka (Kyushu)
Title: Pathologies on log del Pezzo surfaces in characteristic five
Abstract: Log del Pezzo surfaces are 2-dimensional Fano varieties with klt singularities, which form a building block in the minimal model program. Keel-McKernan essentially classified log del Pezzo surfaces over the field of complex numbers around 2000, and Lacini showed that their classification result also holds in characteristic larger than five recently. On the other hand, there are many log del Pezzo surfaces which appear only in low (and positive) characteristics. In this talk, I will explain the classification result on log del Pezzo surfaces which appear only in characteristic five.
Speaker: Hironobu Naoe (Chuo)
Title: Presentation of the fundamental groups of complements of shadows
Abstract: If a 4-manifold collapses onto a simple polyhedron that is locally-flat in the 4-manifold, such a polyhedron is called a shadow of the 4-manifold. Each region of a shadow is equipped with a half-integer called a gleam, and the 4-manifold is reconstructed from the polyhedron and the gleams. In this talk, we focus on a simple polyhedron forming a 2-disk with some annuli attached, which can be considered as a shadow of a 4-ball. Milnor fibers of plane curve singularities and complexified real line arrangements can be represented by such a shadow. We give a presentation of the fundamental group of the complement of a subpolyhedron of such a shadow in the 4-ball. The Wirtinger presentation can also be obtained from our presentation. This is a joint work with Masaharu Ishikawa and Yuya Koda.
Speaker: Takayuki Okuda  (Hiroshima)
Title: Representation theory and combinatorics on compact homogeneous spaces
Abstract: Points arrangements on homogeneous spaces are interested in Algebraic combinatorics. On rank one compact symmetric spaces (ex; spheres or projective spaces), Delsarte theory, an analytical and geometrical method by applying Fourier analysis and Representation theory, were established as strong tools to study finite points arrangements. However, on higher rank compact symmetric spaces or on non-symmetric homogeneous spaces (especially for non-commutative cases), Delsarte theories were formulated only for some special situations.  In this talk, we give an abstract but general formulation of Delsarte theory for finite points arrangements on compact homogeneous spaces.
Speaker: Sakumi Sugawara (Hokkaido)
Title: Divides with cusps and Kirby diagrams for line arrangements
Abstract: It is known that the cell decomposition of the complement of the  complexified real line arrangements is described in terms of its real structure. In this talk, we give explicit handle decomposition. Since the complement of line arrangement is real 4-manifold, its handle decomposition can be expressed by the Kirby diagram. To describe the Kirby diagram, we introduce the notion of divides with cusps which is the generalization of the divide introduced by A’Campo. This is joint work with Masahiko Yoshinaga.
Speaker: Jun Ueki (Tokyo Denki)
Title: Weber’s class number problem for cyclic covers of knots
Abstract: We study an analogue of Weber’s class number problem for cyclic covers of knots in the spirit of arithmetic topology, namely, in a view of the analogy between the class numbers of number fields and the sizes of the 1st homology groups of 3-manifolds. Let $p$ be a prime number. Weber’s problem is a famous unsolved question with a long history asking the class numbers of certain cyclic $p^n$-th extensions of the rationals $\mathbb{Q}$. We instead examine the sizes $r_{p^n}$ of 1st homology groups of cyclic $p^n$-fold covers of knots in $S^3$. Livingston previously determined when $r_{p^n}=1$ holds for all $p$ and $n$. We concern the weak Weber conjecture’’; among other things, we introduce a certain numerical invariant $\lim_{n\to \infty} r_{p^n}$ which is a unit of the ring of p-adic integers $\mathbb{Z}_p=\varprojlim_n \mathbb{Z}/p^n\mathbb{Z}$ and determine the values for torus knots and twist knots. We remark that our work is closely related to the study of profinite rigidity due to Michel Boileau and others. This talk is based on a joint work with Hyuga Yoshizaki at Tokyo University of Science.

Organizers:
Takuro Abe (Kyushu)
Shinzo Bannai (Okayama University of Science)
Benoît Guerville-Ballé (Kyushu)
Naoki Kitazawa (Kyushu)
Osamu Saeki (Kyushu)
Makoto Sakuma (Osaka City)